160 BELL SYSTEM TECHNICAL JOURNAL 



where f{p) denotes \/pH{p). Substitution of the value of //(/), as 

 given by (2), in (3), gives the transform 



f(p) = -L e-,i(U J\z)e''dz. (32) 



IlTl Jo Jc-lx 



In addition, since //(/) = for / < 0, we must have 



• C+ioo 



-^ I f{p)e'Pdp = when / < 0. 



(33) 



Equations (32) and (33) formulate the necessary and sufficient 

 restrictions on f(p) for the existence of a solution of the operational 

 equation 



h = pfip) = l/H{p). 



To correlate the transform (32) more closely with the classical 

 Fourier transform, write p — n -\- ioo and 



f{ii + 7co) = 0(w) i( and co real. 

 Then the transform (32) becomes 



0(co) =— c'""(It (l>{x)e''Hx (34) 



ZrJo J -00 



for all values of u. > c. Also since //(/) = 0, for / < 0, the lower 

 limit of integration with respect to / in {33) may be replaced by — oo , 

 whence 



0( 



w 



= ^ e-^'hll I <p{x)e"'dx, (35: 



27rJ_oo J_oo 



which is the classical Fourier transform. 



The foregoing naturally suggests a few remarks regarding the mode 

 of approach to the operational calculus. If we regard, as Heaviside 

 certainly did, the operational equation as the symbolic formulation 

 of a definite physical problem, it is not permissible to define the sig- 

 nificance of the operator p a priori. The meaning of the operator p 

 and methods of solution of the equation must be so determined as 

 to give the correct solution of the original physical problem. Hea\i- 

 side's procedure here was purely heuristic and "experimental"; equa- 

 tions (2) and (3), however, provide a sound logical basis for the de- 

 velopment of the operational calculus. On the other hand, from the 

 purely mathematical standpoint it is possible to develop an opera- 



